Calculating the dimension of a cotangent space of a curve
$begingroup$
Let $A$ be the local ring
$$A=left(frac{mathbb{C}[x,y]}{(y^2-x^3-x^2)}right)_{(x,y)}$$
$mathfrak{m}=(x,y)A$ be the maximal ideal in $A$. I'm asked to calculate the dimension of the $mathbb{C}$-vector space $mathfrak{m}/mathfrak{m}^2$ which is the dimension of the cotangent space at $(0,0)$. I do it as follows.
$$
begin{aligned}
mathfrak{m}/mathfrak{m}^2&=left(frac{(x,y)}{(y^2-x^3-x^2)}right)_{(x,y)}/left(frac{(x,y)^2}{(y^2-x^3-x^2)}right)_{(x,y)}\
&=left(frac{(x,y)_{(x,y)}}{(y^2-x^3-x^2)_{(x,y)}}right)/left(frac{(x,y)^2_{(x,y)}}{(y^2-x^3-x^2)_{(x,y)}}right)\
&=frac{(x,y)_{(x,y)}}{(x,y)^2_{(x,y)}}=left(frac{(x,y)}{(x,y)^2}right)_{(x,y)}simeqmathbb{C}^2
end{aligned}
$$
This seemingly correct argument gives the desired result 2, but I realize that I didn't use any information about $(y^2-x^3-x^2)$, and it will be the same at any other point on $y^2-x^3-x^2$ which is impossible. I think there must be some problem about my argument, but I cannot find it out. Where am I wrong? What is the correct way to do it? Thanks in advance.
algebraic-geometry commutative-algebra
$endgroup$
add a comment |
$begingroup$
Let $A$ be the local ring
$$A=left(frac{mathbb{C}[x,y]}{(y^2-x^3-x^2)}right)_{(x,y)}$$
$mathfrak{m}=(x,y)A$ be the maximal ideal in $A$. I'm asked to calculate the dimension of the $mathbb{C}$-vector space $mathfrak{m}/mathfrak{m}^2$ which is the dimension of the cotangent space at $(0,0)$. I do it as follows.
$$
begin{aligned}
mathfrak{m}/mathfrak{m}^2&=left(frac{(x,y)}{(y^2-x^3-x^2)}right)_{(x,y)}/left(frac{(x,y)^2}{(y^2-x^3-x^2)}right)_{(x,y)}\
&=left(frac{(x,y)_{(x,y)}}{(y^2-x^3-x^2)_{(x,y)}}right)/left(frac{(x,y)^2_{(x,y)}}{(y^2-x^3-x^2)_{(x,y)}}right)\
&=frac{(x,y)_{(x,y)}}{(x,y)^2_{(x,y)}}=left(frac{(x,y)}{(x,y)^2}right)_{(x,y)}simeqmathbb{C}^2
end{aligned}
$$
This seemingly correct argument gives the desired result 2, but I realize that I didn't use any information about $(y^2-x^3-x^2)$, and it will be the same at any other point on $y^2-x^3-x^2$ which is impossible. I think there must be some problem about my argument, but I cannot find it out. Where am I wrong? What is the correct way to do it? Thanks in advance.
algebraic-geometry commutative-algebra
$endgroup$
$begingroup$
You might find my answer here useful.
$endgroup$
– André 3000
Dec 13 '18 at 5:03
1
$begingroup$
@André 3000 It helps, thanks.
$endgroup$
– Y.Guo
Dec 13 '18 at 5:26
$begingroup$
Actually, I just realized that this question deals with the same ring as in your question.
$endgroup$
– André 3000
Dec 13 '18 at 5:28
$begingroup$
@Y.Guo Please explain why "it will be the same at any other point on $y^2-x^3-x^2$ which is impossible".
$endgroup$
– user26857
Dec 13 '18 at 22:06
add a comment |
$begingroup$
Let $A$ be the local ring
$$A=left(frac{mathbb{C}[x,y]}{(y^2-x^3-x^2)}right)_{(x,y)}$$
$mathfrak{m}=(x,y)A$ be the maximal ideal in $A$. I'm asked to calculate the dimension of the $mathbb{C}$-vector space $mathfrak{m}/mathfrak{m}^2$ which is the dimension of the cotangent space at $(0,0)$. I do it as follows.
$$
begin{aligned}
mathfrak{m}/mathfrak{m}^2&=left(frac{(x,y)}{(y^2-x^3-x^2)}right)_{(x,y)}/left(frac{(x,y)^2}{(y^2-x^3-x^2)}right)_{(x,y)}\
&=left(frac{(x,y)_{(x,y)}}{(y^2-x^3-x^2)_{(x,y)}}right)/left(frac{(x,y)^2_{(x,y)}}{(y^2-x^3-x^2)_{(x,y)}}right)\
&=frac{(x,y)_{(x,y)}}{(x,y)^2_{(x,y)}}=left(frac{(x,y)}{(x,y)^2}right)_{(x,y)}simeqmathbb{C}^2
end{aligned}
$$
This seemingly correct argument gives the desired result 2, but I realize that I didn't use any information about $(y^2-x^3-x^2)$, and it will be the same at any other point on $y^2-x^3-x^2$ which is impossible. I think there must be some problem about my argument, but I cannot find it out. Where am I wrong? What is the correct way to do it? Thanks in advance.
algebraic-geometry commutative-algebra
$endgroup$
Let $A$ be the local ring
$$A=left(frac{mathbb{C}[x,y]}{(y^2-x^3-x^2)}right)_{(x,y)}$$
$mathfrak{m}=(x,y)A$ be the maximal ideal in $A$. I'm asked to calculate the dimension of the $mathbb{C}$-vector space $mathfrak{m}/mathfrak{m}^2$ which is the dimension of the cotangent space at $(0,0)$. I do it as follows.
$$
begin{aligned}
mathfrak{m}/mathfrak{m}^2&=left(frac{(x,y)}{(y^2-x^3-x^2)}right)_{(x,y)}/left(frac{(x,y)^2}{(y^2-x^3-x^2)}right)_{(x,y)}\
&=left(frac{(x,y)_{(x,y)}}{(y^2-x^3-x^2)_{(x,y)}}right)/left(frac{(x,y)^2_{(x,y)}}{(y^2-x^3-x^2)_{(x,y)}}right)\
&=frac{(x,y)_{(x,y)}}{(x,y)^2_{(x,y)}}=left(frac{(x,y)}{(x,y)^2}right)_{(x,y)}simeqmathbb{C}^2
end{aligned}
$$
This seemingly correct argument gives the desired result 2, but I realize that I didn't use any information about $(y^2-x^3-x^2)$, and it will be the same at any other point on $y^2-x^3-x^2$ which is impossible. I think there must be some problem about my argument, but I cannot find it out. Where am I wrong? What is the correct way to do it? Thanks in advance.
algebraic-geometry commutative-algebra
algebraic-geometry commutative-algebra
asked Dec 13 '18 at 3:15
Y.GuoY.Guo
1619
1619
$begingroup$
You might find my answer here useful.
$endgroup$
– André 3000
Dec 13 '18 at 5:03
1
$begingroup$
@André 3000 It helps, thanks.
$endgroup$
– Y.Guo
Dec 13 '18 at 5:26
$begingroup$
Actually, I just realized that this question deals with the same ring as in your question.
$endgroup$
– André 3000
Dec 13 '18 at 5:28
$begingroup$
@Y.Guo Please explain why "it will be the same at any other point on $y^2-x^3-x^2$ which is impossible".
$endgroup$
– user26857
Dec 13 '18 at 22:06
add a comment |
$begingroup$
You might find my answer here useful.
$endgroup$
– André 3000
Dec 13 '18 at 5:03
1
$begingroup$
@André 3000 It helps, thanks.
$endgroup$
– Y.Guo
Dec 13 '18 at 5:26
$begingroup$
Actually, I just realized that this question deals with the same ring as in your question.
$endgroup$
– André 3000
Dec 13 '18 at 5:28
$begingroup$
@Y.Guo Please explain why "it will be the same at any other point on $y^2-x^3-x^2$ which is impossible".
$endgroup$
– user26857
Dec 13 '18 at 22:06
$begingroup$
You might find my answer here useful.
$endgroup$
– André 3000
Dec 13 '18 at 5:03
$begingroup$
You might find my answer here useful.
$endgroup$
– André 3000
Dec 13 '18 at 5:03
1
1
$begingroup$
@André 3000 It helps, thanks.
$endgroup$
– Y.Guo
Dec 13 '18 at 5:26
$begingroup$
@André 3000 It helps, thanks.
$endgroup$
– Y.Guo
Dec 13 '18 at 5:26
$begingroup$
Actually, I just realized that this question deals with the same ring as in your question.
$endgroup$
– André 3000
Dec 13 '18 at 5:28
$begingroup$
Actually, I just realized that this question deals with the same ring as in your question.
$endgroup$
– André 3000
Dec 13 '18 at 5:28
$begingroup$
@Y.Guo Please explain why "it will be the same at any other point on $y^2-x^3-x^2$ which is impossible".
$endgroup$
– user26857
Dec 13 '18 at 22:06
$begingroup$
@Y.Guo Please explain why "it will be the same at any other point on $y^2-x^3-x^2$ which is impossible".
$endgroup$
– user26857
Dec 13 '18 at 22:06
add a comment |
1 Answer
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$begingroup$
I’ll be really sloppy here; you can clean up after me.
Consider the nonsingular point $P=(-1,0)$, so that you want to localize at $(x+1,y)$. Your generators seem to be $x+1$ and $y$, but since $x$ is a unit in the localization, you could generate the maximal ideal just as well by $(x+1)x^2$ and $y$. In other words, by $y^2$ and $y$, thus by $y$ alone.
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add a comment |
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1 Answer
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1 Answer
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active
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active
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active
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votes
$begingroup$
I’ll be really sloppy here; you can clean up after me.
Consider the nonsingular point $P=(-1,0)$, so that you want to localize at $(x+1,y)$. Your generators seem to be $x+1$ and $y$, but since $x$ is a unit in the localization, you could generate the maximal ideal just as well by $(x+1)x^2$ and $y$. In other words, by $y^2$ and $y$, thus by $y$ alone.
$endgroup$
add a comment |
$begingroup$
I’ll be really sloppy here; you can clean up after me.
Consider the nonsingular point $P=(-1,0)$, so that you want to localize at $(x+1,y)$. Your generators seem to be $x+1$ and $y$, but since $x$ is a unit in the localization, you could generate the maximal ideal just as well by $(x+1)x^2$ and $y$. In other words, by $y^2$ and $y$, thus by $y$ alone.
$endgroup$
add a comment |
$begingroup$
I’ll be really sloppy here; you can clean up after me.
Consider the nonsingular point $P=(-1,0)$, so that you want to localize at $(x+1,y)$. Your generators seem to be $x+1$ and $y$, but since $x$ is a unit in the localization, you could generate the maximal ideal just as well by $(x+1)x^2$ and $y$. In other words, by $y^2$ and $y$, thus by $y$ alone.
$endgroup$
I’ll be really sloppy here; you can clean up after me.
Consider the nonsingular point $P=(-1,0)$, so that you want to localize at $(x+1,y)$. Your generators seem to be $x+1$ and $y$, but since $x$ is a unit in the localization, you could generate the maximal ideal just as well by $(x+1)x^2$ and $y$. In other words, by $y^2$ and $y$, thus by $y$ alone.
answered Dec 13 '18 at 3:39
LubinLubin
44.7k44586
44.7k44586
add a comment |
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$begingroup$
You might find my answer here useful.
$endgroup$
– André 3000
Dec 13 '18 at 5:03
1
$begingroup$
@André 3000 It helps, thanks.
$endgroup$
– Y.Guo
Dec 13 '18 at 5:26
$begingroup$
Actually, I just realized that this question deals with the same ring as in your question.
$endgroup$
– André 3000
Dec 13 '18 at 5:28
$begingroup$
@Y.Guo Please explain why "it will be the same at any other point on $y^2-x^3-x^2$ which is impossible".
$endgroup$
– user26857
Dec 13 '18 at 22:06